Enhanced charge density wave with mobile superconducting vortices in La1.885Sr0.115CuO4

Superconductivity in the cuprates is found to be intertwined with charge and spin density waves. Determining the interactions between the different types of order is crucial for understanding these important materials. Here, we elucidate the role of the charge density wave (CDW) in the prototypical cuprate La1.885Sr0.115CuO4, by studying the effects of large magnetic fields (H) up to 24 Tesla. At low temperatures (T), the observed CDW peaks reveal two distinct regions in the material: a majority phase with short-range CDW coexisting with superconductivity, and a minority phase with longer-range CDW coexisting with static spin density wave (SDW). With increasing magnetic field, the CDW first grows smoothly in a manner similar to the SDW. However, at high fields we discover a sudden increase in the CDW amplitude upon entering the vortex-liquid state. Our results signify strong coupling of the CDW to mobile superconducting vortices and link enhanced CDW amplitude with local superconducting pairing across the H − T phase diagram.

show the 20 zero-field measurements taken before and after the field pulse, and the red solid vertical line represents the x-ray pulse arrives at the peak of the magnetic field.

II. SAMPLE CHARACTERIZATION
The superconducting transition temperature (T c ) of the La 1.885 Sr 0.115 CuO 4 (LSCO) sample was determined by magnetic susceptibility using a Physical Properties Measurement System from Quantum Design, Inc. The real part of the magnetic susceptibility, χ , measured with 4 Oe AC magnetic field at 1000 Hz, is shown in Supplementary Fig. 2.

III. NUCLEAR BRAGG PEAKS AND STRUCTURAL TWIN DOMAINS
The orientation of the sample was determined by measuring the (204) and (206) nuclear Bragg peaks, which is then used to convert the data from the detector pixel coordinate to the reciprocal space coordinate of the sample. Supplementary Figs. 3a-c show the h−, k−, and l− cuts through the (206) Bragg peak measured at 28 K. The peak widths as determined by fitting to these cuts are used to estimate the instrumental resolution at the nearby (2.235,0,5.5) CDW peak position.
LSCO sample generally develops four structural twin domains below the tetragonalorthorhombic structural transition temperature (∼ 250 K for the doping level studied here 1 ), which can be considered as two pairs of twin domains that are related by 90 • rotation 2 . As a result, we expect to see three (206) peaks that are split along the k-direction: a central peak due to one pair of domains, and two side peaks due to the other pair. As shown in Supplementary Fig. 3b, the two side peaks are much weaker compared to the central peak (∼7 % as estimated by ratio of the integrated peak intensities), suggesting that our LSCO sample is dominated by just one pair of twin domains, similar to the situation observed in a previous study 2 .

TIONS
The total scattering intensity of the CDW peak, defined as the sum of the scattering intensities within the dashed rectangle indicated in Supplementary Fig. 1a, measured as a function of the sample rotation angle (θ) at 28 K is shown in Supplementary Fig. 4. For high-field measurements the sample rotation angle is fixed at the peak center of θ = 30.8 • . We can take a closer look at the CDW spatial correlations by projecting the CDW θscan data onto reciprocal space coordinate. As shown in Fig  δ k /2q cdw = 2.5(1) • at 6.5 K, however, is an order of magnitude larger than what one expects from simple structural twinning effect 3 , and suggests that the CDW does not propagate along the Cu-Cu bond direction, consistent with previous soft x-ray scattering measurements 3,4 . θ Y of 2.5(1) • is similar to that for the SDW order (∼ 3 • ) in La 1.88 Sr 0.12 CuO 4 2 , providing further support for the inter-locking between CDW and SDW orders as in the spin-charge stripe order 5,6 . The splitting persists at high temperatures in the absence of the SDW order Supplementary Fig. 6b]. This suggests a common origin for θ Y in CDW SRO and CDW stripe , such as anisotropy in the second-neighbor electron hopping in the underlying Hamiltonian 7 .

V. CDW TEMPERATURE DEPENDENCE FOR LSCO IN PRIOR STUDIES
In Supplementary Fig. 7 we summarize the temperature dependence of CDW intensity and peak width for La 1.88 Sr 0.12 CuO 4 reported in prior studies 3,8,9 . Consistent with our data, these data all exhibit contradictory behavior between the CDW intensity and the peak Schematic illustration of how two CDW peaks arise from two orthorhombic structural domains.
Error bars represent one standard deviation.
width. The intensity depresses/saturates below T c , while the peak width keeps decreasing monotonically.

TWO-COMPONENT FITTINGS
We can judge the goodness of fit for the single-peak and two-component fittings by comparing reduced χ 2 . As shown in Supplementary Fig. 8, the two-component fittings are consistently better than the single-peak fittings. Note that after fixing the peak width for CDW stripe and CDW SRO , and using a common peak center for these two components, the number of fitting parameters is the same in the two-component fitting as that in the single-peak fitting. is consistent with our starting assumption that the CDW is enhanced in the stripe ordered regions while suppressed in the superconductivity-dominant regions.
It is also interesting to compare the CDW intensities in La 1.885 Sr 0.115 CuO 4 to that in La 1.875 Ba 0.125 CuO 4 (LBCO), a prototypical stripe ordered cuprate where the stripe order is expected to be strongest and the magnetic volume is estimated to be ∼ 100% 10 . In a previous study it was found that the CDW intensity in LSCO is ∼ 1/4 of that in LBCO 3 . Focusing on the stripe ordered regions, the unit volume CDW stripe intensity in LSCO compared to that in LBCO is ∼ {1/[0.18 + (0.82/1.4)]}/4 ∼ 0.3, consistent with a stronger stripe order in LBCO.

IN MAGNETIC FIELDS
Here we elaborate on the main assumption we made to analyze the magnetic field dependence of the CDW in La 1.885 Sr 0.115 CuO 4 , that CDW stripe (H) ∼ SDW(H). First, this is consistent with our finding from the two-component analysis that CDW stripe (T ) ∼ SDW(T ) in zero magnetic field (Fig.2a of the main text). Second, in La 1.88 Sr 0.12 CuO 4 , it is found that at ∼ 7 T, the field-enhanced CDW intensities, which in our model is mostly due to CDW stripe , is proportional to the field-enhanced SDW intensities, as shown in Fig. 2d

IX. ADDITIONAL CHARACTERIZATIONS OF CDW FIELD DEPENDENCE
As described in Section I. Experimental Setup, for each pulsed field measurement the corresponding zero-field data have 20 times more counting time compared to the data collected in field. To take advantage of the higher statistics zero-field data, we fitted the in-field data and the corresponding zero-field data simultaneously, with the constraint that both have the same linear background under the reasonable assumption that the background does not vary with applied magnetic fields, as shown in Supplementary Fig. 10.
In Supplementary Fig. 11 we compare the existing data of field-dependent CDW in La 1.88 Sr 0.12 CuO 4 (up to 10 T) 9 with our measurements. As shown in Supplementary Fig. 11a, the prior data is also well described by CDW stripe , providing additional support for our proposition that in the low field regime, the enhancement to the CDW mostly comes from CDW stripe , while CDW SRO remains nearly unchanged. Assuming that CDW SRO follows a step-like enhancement at H m , and that the peak widths for CDW SRO and CDW stripe remain the same as their zero-field values, we can simulate the field dependence of both intensity and width. As shown in Supplementary Fig. 11, such a simple model provides a semi-quantitative description of the prior data and our data.

X. COMPARISON TO CDW FIELD DEPENDENCE IN YBCO
CDW SRO in LSCO is similar to the short-range CDW (∼ half-integer l) observed in YBCO in that both are suppressed below the superconducting transition T c 14 . In YBCO it is found that the CDW intensities increase linearly with applied magnetic fields in the lowfield regime 15 . If there is no spin-charge stripe order, we would expect CDW SRO in LSCO to increase with applied field even at low fields, like YBCO. However, the stripe order in LSCO is also enhanced by applied magnetic fields, and according to µSR measurements this enhancement is mostly due to increase in volume fraction of the stripe ordered regions 16 .
Correspondingly, the volume fraction of the CDW SRO region is reduced. Therefore, the magnetic-field-induced enhancement to CDW SRO , due to weakening of superconductivity, is offset by the reduction in the volume fraction. Our data and analysis suggests that the net effect is a weak field dependence of CDW SRO intensity in the low-field regime. As such, the CDW SRO behavior in LSCO may be seen as consistent with the short-range CDW in YBCO, and it is the rapid reduction in CDW SRO volume fraction in the low-field regime that renders a weak field dependence for CDW SRO in LSCO.
At larger fields, we find a large enhancement to the CDW SRO intensities upon entering the vortex-liquid state in LSCO. A close inspection of the field dependence of the short-range CDW in YBCO (reproduced in Supplementary Figs. 12a-b using data in ref. 18) across the corresponding vortex-melting field (H m ) also suggests a CDW anomaly around H m . As shown in Supplementary Fig. 12a, the linear CDW intensity enhancement at low fields turns into a plateau above H m , a field that is only ∼ 60% of H c2 17 . The CDW correlation length shown in Supplementary Fig. 12b is also consistent with an anomaly ∼ H m . Together with our data, this suggests that a strong response of short-range CDW to the vortex-melting transition could be a common phenomenon in cuprate superconductors.
It is found in YBCO superconductors that high magnetic field induces a three-dimensional CDW peak at integer l positions 18,23,24 . As shown in Fig. 3a of the main text, there is no apparent magnetic-field-enhanced CDW intensities at l = 6 at 24 T in LSCO. Systematic h−cuts at l = 6 at various fields, as shown in Supplementary Fig. 13, indicate the absence of CDW intensity at l = 6 up to at least 24 T.